Is a charity given in Ramadan with the intention of aid, and not with the intention of it being Zakat al-Fitr, sufficient as Zakat al-Fitr, even though the giver did not intend it to be Zakat al-Fitr?

Zakat al-Fitr is an obligatory duty. One who intentionally neglects it while being able to, is sinful. It is not valid without the intention. Whatever a person gives out in charity during Ramadan as aid to the needy does not suffice for Zakat al-Fitr due to the absence of intention. However, if one intended it as Zakat al-Fitr, it is valid according to the madhhabs of Abu Hanifa and Shafi'i. It is not valid according to the madhhabs of Malik and Ahmad unless it was given out one or two days before Eid, and this is closer to the evidence. Delaying it beyond the day of Eid is Haram by consensus, and one's obligation is not discharged except by giving it out as a make-up (qada').